The spatial string tension for 3+1 dimensional QCD at finite temperature is measured. The gauge configurations we re generated with two light and one heavier strange quark on lattices of size $16^3 4$ and $24^3 6$. This spatial string tension is compared with the string tension of the 3 dimensional pure gauge theory together with the temperature dependent 2-loop running coupling. Further comparison is made with predictions from dimensionally reduced effective theories.
We study the string tension as a function of temperature, fitting the SU(3) lattice QCD finite temperature free energy potentials computed by the Bielefeld group. We compare the string tension points with order parameter curves of ferromagnets, superconductors or string models, all related to confinement. We also compare the SU(3) string tension with the one of SU(2) Lattice QCD. With the curve providing the best fit to the finite temperature string tensions, the spontaneous magnetization curve, we then show how to include finite temperature, in the state of the art confining and chiral invari…
Spatial string tension is computed in finite temperature gluodynamics on asymmetric lattices in a spherical model approximation. Conditions of scaling behavior are specified. Discrepancies with a standard renormalisation procedure are discussed.
A review article on string tension concept and their relevance as non-perturbative quantity on the study of quark confinement in lattice gauge theories. A detailed description of a variety of methods to measure the string tension on the lattice and an indication of the most promising developments is proposed.
Monopole and photon contributions to abelian Wilson loops are calculated using Monte-Carlo simulations of finite-temperature $SU(2)$ QCD in the maximally abelian gauge. The string tension is reproduced by monopole contribution alone also in finite temperature SU(2) QCD. The spatial string tension scales as $\sqrtσ \propto g^{2}(T)T$ and is reproduced almost by monopole contribution alone. Each configuration has one long monopole loop, and the long monopole loops alone are responsible for the string tension in the confinement phase. On the other hand, the spatial string tension in the deconfine…
Monopole and photon contributions to abelian Wilson loops are calculated using Monte-Carlo simulations of finite-temperature $SU(2)$ QCD in the maximally abelian gauge. Long monopole loops alone are responsible for the behavior of the string tension in the confinement phase up to the critical $β_c$. Short monopole loops and photons do not contribute to the string tension. The abelian and the monopole spacial string tensions (both of which agree with the normal ones for $β< β_c$) show a $g^{4}(T) T^2$ scaling behavior in the deconfinement phase. The abelian spacial string tension is in agreemen…
We investigate Abelian and monopole contributions to spatial string tension in the deconfined phase of finite temperature SU(2) gauge theory without imposing any gauge fixing conditions. Lattice calculations of non-Abelian and Abelian spatial string tensions from the Wilson action at gauge coupling $β=2.74$ and lattice volume $24^3\times{N_t}$ $(N_{t}=\left\{ 24,8,6,4,2)\right\}$ show that these string tensions agree with each other within error bars at any adopted value of $N_t$, which implies Abelian dominance. From measurements of non-Abelian, Abelian, and monopole forces that arise from th…
We calculate the spatial string tension in (2+1) flavor QCD with physical strange quark mass and almost physical light quark masses using lattices with temporal extent N_tau=4,6 and 8. We compare our results on the spatial string tension with predictions of dimensionally reduced QCD. This suggests that also in the presence of light dynamical quarks dimensional reduction works well down to temperatures 1.5T_c.
In response to recently renewed interests in SU(N) pure-gauge dynamics with large N, both from M/string duality and from finite-temperature QCD phase structure, we calculate string tensions acting between the fundamental 4, diquark 6 and other color charges in SU(4) pure-gauge theory at temperatures below the deconfining phase change and above the bulk phase transition. Our results suggest 4 and 6 representations have different string tensions, with a ratio of about 1.3. We also found the deconfining phase change is not strong.
We study the distribution of the color fields due to a static quark-antiquark pair in SU(2) lattice gauge theory. We find evidence of dual Meissner effect. We put out a simple relation between the penetration length and the string tension.
The critical curve of string tension as a function of the temperature is computed in SU(3) Lattice QCD. We present the results for the string tension utilizing a pair of Polyakov loop and antiloop, with two different techniques. We compare the colour averaged loop-antiloop which is gauge invariant but is only adequate to study the string tension, and the colour singlet loop-antiloop using the Landau gauge fixing which also enables to compute the coulomb part of the free energy.
Wilson loops in the adjoint representation are evaluated on cooled lattices in SU(2) lattice gauge theory. It is found that the string tension of an adjoint Wilson loop vanishes, if the loop is evaluated in a sub-ensemble of configurations in which no center vortex links the loop. This result supports our recent proposal that the adjoint string tension, in the Casimir-scaling regime, can be attributed to a center vortex mechanism.
We apply smoothing to SU(2) lattice field configurations in 3+1 dimensions before fixing to the maximally Abelian gauge. The Abelian projected string tension is shown to be stable under this, whilst the monopole string tension declines by O(30%). Blocking of the SU(2) fields reduces this effect, but the use of extended monopole definitions does not. We discuss these results in the context of additional confining excitations in the U(1) vacuum.
In pure SU(N) gauge theory in four dimensions, we determine the string tension at large N from smeared rectangular Wilson loops on the lattice. We learn how well loops of sizes barely on the strong-coupling side of the large-N transition in their eigenvalue distribution can be described by effective string theory.
The spatial string tension across a crossover from the low temperature phase to the high temperature phase is computed in QCD with two flavors of non-perturbatively improved Wilson fermions at small lattice spacing a \sim 0.12fm. We find that in the low temperature phase spatial string tension agrees well with zero temperature string tension. Furthermore, it does not show increasing for temperatures up to T = 1.36 T_{pc}, the highest temperature considered. Our results agree with some theoretical predictions.
A lower bound on the string tension for large beta in SU(2) LGT is derived. The derivation is from first principles and bounds the string tension from below by the expectation for the excitation of a single `tagged' thick vortex winding around the lattice. Thus confinement follows if this expectation remains nonvanishing at large beta. Numerical simulations are presented to show that this is indeed the case.
We report a method to select optimal smearing parameters before production runs and discuss the advantages of this selection for the determination of the string tension.
The spatial string tension, a classic non-perturbative probe for the convergence of the weak-coupling expansion at high temperatures, can be determined in full QCD as well as in a dimensionally reduced effective theory. Comparing both approaches, we find surprisingly good agreement almost down to the critical temperature of the deconfinement phase transition.
We study the perfect monopole action as an infrared effective theory of SU(2) QCD. It is transformed exactly into a lattice string model. Since the monopole interactions are weak in the infrared SU(2) QCD, the string interactions become strong. The strong coupling expansion of string model shows the quantum fluctuation is small. The classical string tension is estimated analytically, and we see it is very close to the quantum one in the SU(2) QCD. We also discuss how to calculate the glueball mass in our model.
We compute, for the four-dimensional SU(4) and SU(6) gauge theories formulated on a lattice, the string tensions sigma_k related to sources with Z_N charge k, using Monte Carlo simulations. Our results are compatible with sigma_k \propto sin (k pi/N), and show sizeable deviations from Casimir scaling.
We present a method for locating center vortices (``fluxons'') in thermalized lattice gauge field configurations. We find evidence, in lattice Monte Carlo simulations, that the asymptotic string tension of fundamental-representation Wilson loops is due to fluctuations in the number of center vortices linking those loops.
We consider whether the 1/N corrections to k-string tensions must begin at order 1/N^2, as in the Sine Law, or whether odd powers of 1/N, as in Casimir Scaling, are also acceptable. The issue is important because different models of confinement differ in their predictions for the representation-dependence of k-string tensions, and corrections involving odd powers of 1/N would seem to be ruled out by the large-N expansion. We show, however, that k-string tensions may, in fact, have leading 1/N corrections, and consistency with the large-N expansion, in the open string sector, is achieved by an …
We numerically compute the string tension in the large N limit of three dimensional Yang-Mills theory using Wilson loops. Space-time Wilson loops are formed using smeared space-like links and unsmeared time-like links. We use partial reduction and both unfolded and folded Wilson loops in the analysis.
We calculate the low-lying glueball spectrum and various string tensions in SU(N) lattice gauge theories in 2+1 dimensions, and extrapolate the results to the continuum limit. We do so for for the range N=2 to N=16 so as to control the N-dependence with a useful precision. We observe a number of striking near-degeneracies in the various J^PC sectors of the glueball spectrum, in particular between C=+ and C=- states. We calculate the string tensions of flux tubes in a number of representations, and provide evidence that the leading correction to the N-dependence of the k-string tensions varies …
We studied a behavior of monopole currents in the high temperature (deconfinement) phase of abelian projected finite temperature SU(2) QCD in maximally abelian gauge. Wrapped monopole currents closed by periodic boundary play an important role for the spatial string tension which is a non-perturbative quantity in the deconfinement phase. The wrapped monopole current density seems to be non-vanishing in the continuum limit. These results may be related to Polyakov's analysis of the confinement mechanism using monopole gas in 3-dimensional SU(2) gauge theory with Higgs fields.
I report on the status of a computation of fundamental and some higher representation string tensions in pure gauge SU(3). An order a^2 tadpole improved action and an anisotropic lattice are used. At present, the static quark potentials and the string tensions are calculated by measuring Wilson loops on an 8^3 x 24 lattice. Wilson loops for higher representations are measured in terms of Wilson loops of the fundamental representation. At the small and intermediate distances available, rough agreement with Casimir scaling is observed, and no color screening for the 8 representation is seen.
We investigate the features of ${\rm CP}^{N-1}$ models concerning confinement and topology. In order to study the approach to the large-$N$ asymptotic regime, we determine the topological susceptibility and the string tension for a wide range of values of $N$, in particular $N=4,10,21,41$. Quantitative agreement with the large-$N$ predictions is found for the ${\rm CP}^{20}$ and the ${\rm CP}^{40}$ models. Problems related to the measure of the topological susceptibility and the string tension on the lattice are discussed.
We discuss a new method for testing confinement and measuring the string tension (in the Coulomb gauge). Our numerical simulations demonstrate that the problems related to Gribov copies are not harmful and that the method is effective in the case of pure gauge Q.C.D.. We discuss the relevance of the method for studying gauge theories coupled to fermionic matter.
A center vortex model for the infrared sector of SU(4) Yang-Mills theory is constructed such as to reproduce both the ratio between the zero-temperature quark and diquark string tensions known from lattice Yang-Mills theory, as well as the properties of the deconfinement transition. On this basis, the temperature dependence of the spatial quark and diquark string tensions is predicted. Though still phenomenologically viable, details of the construction of the SU(4) center vortex model corroborate previous arguments that modeling infrared Yang-Mills dynamics purely in terms of vortex world-surf…
Compact U(1) lattice gauge theory in (2+1) dimensions is studied on anisotropic lattices using Standard Path Integral Monte Carlo techniques. We extract the static quark potential and the string tension from 1.0 <= Dtau <= 0.333 simulations at 1.0 <= beta <= 3.0. Estimating the actual value of the renormalization constant, (c = 44), we observe the evidence of scaling in the string tension for 1.4142 <= beta <= 2.5; with the asymptotic behaviour in the large-beta limit given by K sqrt(beta) = e^(-2.494 beta +2.29). Extrapolations are made to the extreme anisotropic or "Hamiltonian" limit, and c…
We present results for the confinement-deconfinement interface tension $σ_{cd}$ of quenched QCD. They were obtained by applying Binder's histogram method to lattices of size $L^2\times L_z\times L_t$ for $L_t=2$ and $L=8,10,12\mbox{ and }14$ and various $L_z\in [L,\, 4\, L]$. The use of a multicanonical algorithm and rectangular geometries have turned out to be crucial for the numerical studies. We also give an estimate for $σ_{cd}$ at $L_t=4$ using published data.
We present results for the confinement-deconfinement interface tension $α_{cd}$ of quenched QCD. They were obtained by applying Binder's histogram method to lattices of size $L^2\times L_z\times L_t$ for $L_t=2$ and $L=8,10,12\mbox{ and }14$ with $L_z=30$ for $L=8$ and $L_z=3L$ otherwise. The use of a multicanonical algorithm and cylindrical geometries have turned out to be crucial for the numerical studies.
We compute, via numerical simulations, the non-perturbative Coulomb potential of pure SU(3) gauge theory in Coulomb gauge. We find that that the Coulomb potential scales nicely in accordance with asymptotic freedom, that the Coulomb potential is linear in the infrared, and that the Coulomb string tension is about four times larger than the asymptotic string tension. We explain how it is possible that the asymptotic string tension can be lower than the Coulomb string tension by a factor of four.
We present simulation results comparing the SU(3) heavy quark potential extracted from the full Wilson loop expectation to that extracted from the expectation of the Wilson loop fluctuation solely by elements of Z(3). The two potentials are found to coincide. This agreement is stable under multiple smoothings of the configurations which remove short distance fluctuations, and thus reflects long-distance physics. It strongly indicates that the asymptotic string tension arises from thick center vortices linking with the Wilson loop.
Lattice simulations are presented showing the expectation of the fluctuation of the Wilson loop solely by elements of the center to fully reproduce the SU(3) heavy quark potential. The results are stable under smoothing, and point to thick vortices as being responsible for the full SU(3) tension. An analytic result on the necessary presence of thick vortices for confinement at weak coupling is also presented.
We establish a close relation between the spatial string tension of the (3+1)-dimensional $SU(3)$ gauge theory at finite temperature ($σ_s$) and the string tension of the 3-dimensional $SU(3)$ gauge theory ($σ_3$) which is similar to what has been found previously for $SU(2)$. We obtain $\sqrt{σ_3} = (0.554 \pm 0.004) g_3^2$ and $\sqrt{σ_s} = (0.586 \pm 0.045)g^2(T) T$, respectively. For temperatures larger than twice the critical temperature results are consistent with a temperature dependent coupling running according to the two-loop $β$-function with $Λ_T = 0.118(36)T_c$.
We study the spatial volume dependence of electric flux energies for SU(2) Yang-Mills fields on the torus with twisted boundary conditions. The results approach smoothly the rotational invariant Confinement regime. The would-be string tension is very close to the infinite volume result already for volumes of $(1.2 \ {\rm fm.})^3$. We speculate on the consequences of our result for the Confinement mechanism.
We study the properties of the color-Coulomb string tension obtained from the instantaneous part of gluon propagators in Coulomb gauge using quenched SU(3) lattice simulation. In the confinement phase, the dependence of the color-Coulomb string tension on the QCD coupling constant is smaller than that of the Wilson loop string tension. On the other hand, in the deconfinement phase, the color-Coulomb string tension does not vanish even for $T/T_c = 1 \sim 5$, the temperature dependence of which is comparable with the magnetic scaling, dominating the high temperature QCD. Thus, the color-Coulomb…
We numerically compute the spatial string tension in the deconfined phase of three dimensional QCD in the large N limit. Our results clearly show that the string tension grows linearly with temperature.
Global string networks may be relevant in axion production in the early Universe, as well as other cosmological scenarios. Such networks contain a large hierarchy of scales between the string core scale and the Hubble scale, log(f/H) around 70, which influences the network dynamics by giving the strings large tensions T = pi f^2 log(f/H). We present a new numerical approach to simulate such global string networks, capturing the tension without an exponentially large lattice.
We consider three-dimensional SU(N) gauge theories with massless Dirac or Majorana fermions in the adjoint representation. We use the Eguchi-Kawai volume reduction to two-dimensions to calculate the tension sigma_k of the k-string in such theories. Under some assumptions whose validity we discuss, we derive the previously conjectured sine formula, sigma_k ~ N sin pi k/N.
Continuum reduction and Monte Carlo simulation are used to calculate the heavy quark potential and the string tension in large N Yang-Mills theory in four dimensions. The potential is calculated out to a separation of nine lattice units on a lattice with extent six in each direction.
We present results of modeling the temperature dependence of the spatial string tension and thermal phase transition in a five-dimensional framework nowadays known as AdS/QCD. For temperatures close to the critical one we find a behaviour remarkably consistent with the lattice results.
We investigate the properties of global cosmic string networks as a function of the ratio of string tension to Goldstone-field coupling, and as a function of the Hubble damping strength. Our results show unambiguously that the string density is sensitive to this ratio. We also find that existing semi-analytical (one-scale) models must be missing some important aspect of the network dynamics. Our results point the way towards improving such models.
Using smeared Creutz ratios we extract the string tension for SU(N) pure gauge theory and $N$=3,4,5,6,8. We employ these results to extrapolate to large N. The same methodology is applied to the single-site Twisted Eguchi Kawai model. The corresponding string tension matches perfectly within errors with the extrapolated one, providing strong evidence in favour of the twisted reduction framework. Interesting results are also obtained on the behaviour of Creutz ratios for large sizes.
Monopole and photon contributions to abelian Wilson loops are calculated using Monte-Carlo simulations of SU(2) QCD in the maximally abelian gauge. The string tension is well reproduced only by monopole contributions, whereas photons alone are responsible for the Coulomb coefficient of the abelian static potential.
In the low temperature confining phase of QCD or QCD-like theories it is challenging to capture the temperature dependence of observables through AdS/CFT. Using the blackfold approach we compute the quark-antiquark linear static potential in the low temperature confining phase, taking into account the thermal excitations of the string. We find the explicit temperature dependence of the string tension and notice that, as naturally expected, tension decreases as temperature increases. We have also generalized the blackfold approach for the computation of the Wilson loops, making it directly appl…
Recently, we proposed a new method to extract the string tension from 4-dimensionally smeared Wilson loops. In this talk, we first show that the results obtained using this smearing method are identical to those obtained by Wilson flow, once the time step is sufficiently small. We then demonstrate the practical advantage of our method by applying it to the calculation of string tension in SU(3) Yang-Mills theory.
From continuum studies it is known that the Coulomb string tension $σ_C$ gives an upper bound for the physical (Wilson) string tension $σ_W$ [D. Zwanziger, Phys. Rev. Lett. 90, 102001 (2003)]. How does however such relationship translate to the lattice? In this paper we give evidence that there, while the two string tensions are related at zero temperature, they decouple at finite temperature. More precisely, we show that on the lattice the Coulomb gauge confinement scenario is always tied to the spatial string tension, which is known to survive the deconfinement phase transition and to cause …
We explore the performance of CUDA in performing Landau gauge fixing in Lattice QCD, using the steepest descent method with Fourier acceleration. The code performance was tested in a Tesla C2070, Fermi architecture. We also present a study of the string tension at finite temperature in the confined phase. The string tension is extracted from the color averaged free energy and from the color singlet using Landau gauge fixing.
SU(N) gauge theories, extended with adjoint fermions having periodic boundary conditions, are confining at high temperature for sufficiently light fermion mass $m$. In the high temperature confining region, the one-loop effective potential for Polyakov loops has a Z(N)-symmetric confining minimum. String tensions associated with Polyakov loops are calculable in perturbation theory, and display a novel scaling behavior in which higher representations have smaller string tensions than the fundamental representation. In the magnetic sector, the Polyakov loop plays a role similar to a Higgs field,…
We numerically compute the string tension in the large N limit of three dimensional Yang-Mills theory using Wilson loops. Space-time loops are formed as products of smeared space-like links and unsmeared time-like links. We use continuum reduction and both unfolded and folded Wilson loops in the analysis.
We calculate the k=2 string tensions in SU(4) and SU(5) gauge theories in 3+1 dimensions, and compare them to the k=1 fundamental string tensions. We find, from the continuum extrapolation of our lattice calculations, that K(k=2)/K(k=1) = 1.40(8) in the SU(4) gauge theory, and that K(k=2)/K(k=1) = 1.56(10) in SU(5). We remark upon the way this might constrain the dynamics of confinement and the intriguing implications it might have for the mass spectrum of SU(N) gauge theories. We also note that these results agree closely with the MQCD-inspired conjecture that the SU(N) string tensions satisf…
The leading non-perturbative contribution to the static QCD potential at r << 1/Lambda_QCD is known to be O(r^2) in operator-product expansion. It indicates that a "Coulomb+linear" potential at r <~ 1/Lambda_QCD is included in the perturbative QCD prediction of the potential. It was shown that this is indeed the case, and the "Coulomb+linear" potential has been systematically computed up to next-to-next-to-leading logarithmic order, in terms of the Lambda parameter in the MSbar scheme (Lambda_MSbar). We review the present status of the perturbative prediction for the QCD potential, which takes…
It is shown that a Coulomb potential using a running coupling slightly modified from the perturbative form can produce an interquark potential that appears nearly linear over a large distance range. Recent high-statistics SU(2) lattice gauge theory data fit well to this potential without the need for a linear string-tension term. This calls into question the accuracy of string tension measurements which are based on the assumption of a constant coefficient for the Coulomb term. It also opens up the possibility of obtaining an effectively confining potential from gluon exchange alone.
We exhibit the appropriate variables allowing the plus-minus (Z(2)) `reduction' of the Wilson loop operator which provides a direct measure of the thin, thick and `mixed' Z(2) topological, gauge-invariant vortices in SU(2) LGT. Simulations with the Wilson action, as well as a perfect action smoothing procedure, show the string tension to be reproduced from the contributions of these excitations.
The spatial string tension of a SU(N) gauge theory without quarks is calculated using gauge/string duality for 1.2T_c<T<3T_c. The result is remarkably consistent with the available lattice data for N=2,3. Some evidence is found that to leading order the large distance physics of a string can be described by a five-dimensional geometry without a need for an internal space.
We improve upon recent calculations of the low-lying `glueball' spectra of SO(3) and SO(4) lattice gauge theories in 2+1 dimensions, and compare the resulting continuum extrapolations with SU(2). We find that these are reasonably consistent, as are the SU(2) and SO(4) string tensions when these are corrected for the differing representations of the flux. All this indicates that the different global properties of these groups do not play a significant role in the low-lying physics.
We present an analytical continuum calculation, starting from first principles, of the vacuum wavefunction and string tension for pure Yang-Mills theories in $(2+1)$ dimensions, extending our previous analysis using gauge-invariant matrix variables. The vacuum wavefunction is consistent with what is expected at both high and low momentum regimes. The value of the string tension is in very good agreement with recent lattice Monte Carlo evaluations.
We address the question of whether the large-N expansion in pure SU(N) gauge theories requires that k-string tensions must have a power series expansion in 1/N^2, as in the sine law, or whether 1/N contributions are also allowable, as in Casimir scaling. We find that k-string tensions may, in fact, have 1/N corrections, and consistency with the large-N expansion in the open-string sector depends crucially on an exact cancellation, which we will prove, among terms involving odd powers of 1/N in particular combinations of Wilson loops. It is shown how these cancellations are fulfilled, and consi…
